#50455

order	256
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,4)(3,7,8,6)(10,14)(15,18)(16,21)(22,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(10,14)(11,17)(12,20)(22,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^8|a^2=fgh,b^2=f,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#50455, #7705915, #7706225, …
maximal normal subgroup of	C₂×#50455, #7705915, #7706225, …
maximal quotient of	#560760, #561128, #561848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43443);